Three main parts of generalized cell mapping are improved for global analysis. A simple method, which is not based on the theory of digraphs, is presented to locate complete self-cycling sets that corre- spond to attractors and unstable invariant sets involving saddle, unstable periodic orbit and chaotic saddle. Refinement for complete self-cycling sets is developed to locate attractors and unstable in- variant sets with high degree of accuracy, which can start with a coarse cell structure. A nonuniformly interior-and-boundary sampling technique is used to make the refinement robust. For homeomorphic dissipative dynamical systems, a controlled boundary sampling technique is presented to make gen- eralized cell mapping method with refinement extremely accurate to obtain invariant sets. Recursive laws of group absorption probability and expected absorption time are introduced into generalized cell mapping, and then an optimal order for quantitative analysis of transient cells is established, which leads to the minimal computational work. The improved method is applied to four examples to show its effectiveness in global analysis of dynamical systems.
This paper demonstrates and analyses double heteroclinic tangency in a three-well potential model, which can produce three new types of bifurcations of basin boundaries including from smooth to Wada basin boundaries, from fractal to Wada basin boundaries in which no changes of accessible periodic orbits happen, and from Wada to Wada basin boundaries. In a model of mechanical oscillator, it shows that a Wada basin boundary can be smooth.
Spatiotemporal multiple coherence resonances for calcium activities induced by weak Gaussian white noise in coupled hepatocytes are studied.It is shown that bi-resonances in hepatocytes are induced by the interplay and competition between noise and coupling of cells,in other words,the cell in network can be excited either by noise or by its neighbour via gap junction which can transfer calcium ions between cells.Furthermore,the intercellular annular calcium waves induced by noise are observed,in which the wave length decreases with noise intensity augmenting but increases monotonically with coupling strength increasing.And for a fixed noise level,there is an optimal coupling strength that makes the coherence resonance reach maximum.
